Computing quantiles for a discrete random variable by applying Definition 7.1 would require inverting the CDF. However, this is a piecewise constant function, featuring jumps at each value of the distribution support, which makes its inversion impossible in general.
Example 7.4 Consider random demand for a spare part, sold in low volumes, over the next time period. There is no inventory at present, and we must determine the purchased quantity, in such a way that the probability of satisfying the whole demand is above a minimal threshold. Assume that randomness in demand can be modeled by the PMF of Table 7.1, and that we would like to meet demand with a probability of 0.85. In the parlance of supply chain management, we should say that our service level is 85%.5 The purchased amount should correspond to the quantile with 0.85 probability level. A look at Table 7.1 shows that there is no value of x such that FX (x) = 0.85. Indeed, the function is not invertible, as illustrated in Fig. 7.8. What would one do in practice? The sensible solution, since x = 3 gives only a 80% service level, is to select x = 4 to meet the required constraint.
What we have done in the example above makes sense: The quantile is related to a decision, and we make it in such a way to stay on the safe side. In fact, Definition 7.1 can be generalized as follows.
DEFINITION 7.2 (Generalized definition of quantiles) Let FX(x) be the CDF of random variable X. Given a probability level α ∈ [0, 1], we define the quantile xα of the distribution as the smallest number xα such that FX ≥ α. Formally
We immediately see that if the CDF is invertible, this definition boils down to the previous one. Indeed, Eq. (7.8) corresponds to the so-called generalized inverse function. The reader is urged to check that applying this definition, we do find the decision we chose in Example 7.4. Unlike percentiles, quantiles of probability distributions have a precise definition that makes perfect sense, as we will see in the applications described later in Section 7.8. Before proceeding with theoretical concepts, it is worth pausing a little and check a remarkable example.
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